相关论文: Non-freeness of parabolic two-generator groups

We initiate a new, computational approach to a classical problem: certifying non-freeness of ($2$-generator, parabolic) M\"{o}bius subgroups of $\mathrm{SL}(2,\mathbb{Q})$. The main tools used are algorithms for Zariski dense groups and…

Let $\alpha \in \mathbb{R}$ and let $$A=\begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix} \ \text{and} \ B_{\alpha} = \begin{bmatrix} 1 & 0 \\ \alpha & 1\end{bmatrix}.$$ The subgroup $G_\alpha$ of $\mathrm{SL}_2(\mathbb{R})$ is a group generated…

In [J. Algebra 452 (2016), 372-389], we characterise when the sequence of free subgroup numbers of a finitely generated virtually free group $\Gamma$ is ultimately periodic modulo a given prime power. Here, we show that, in the remaining…

Let G be an abelian group and let lambda be the smallest rank of any group whose direct sum with a free group is isomorphic to G. If lambda is uncountable, then G has lambda pairwise disjoint, non-free subgroups. There is an example where…

Let $G$ be a group that is relatively hyperbolic with respect to a collection of subgroups $\{H_{\lambda}\}_{\lambda\in \Lambda}$. Suppose that $G$ is given by a finite relative presentation $\mathcal{P}$ with respect to this collection. We…

Let $q\in\mathbb{C}$, let \[a=\begin{pmatrix} 1&0\\1&1\end{pmatrix},\quad b_q=\begin{pmatrix} 1&q\\0&1\end{pmatrix},\] and let $G_q<\mathrm{SL}_2(\mathbb{C})$ be the group generated by $a$ and $b_q$. In this paper, we study the problem of…

The principal character of a representation of the free group of rank two into PSL(2, C) is a triple of complex numbers that determines an irreducible representation uniquely up to conjugacy. It is a central problem in the geometry of…

We find a family of groups generated by a pair of parabolic elements in which every relator must admit a long subword of a specific form. In particular, this collection contains groups in which the number of syllables of any relator is…

Given a linear equation $\mathcal{L}$, a set $A$ of integers is $\mathcal{L}$-free if $A$ does not contain any `non-trivial' solutions to $\mathcal{L}$. This notion incorporates many central topics in combinatorial number theory such as…

We study the freeness problem for subgroups of $\operatorname{SL}_2(\mathbb{C})$ generated by two parabolic matrices. For $q = r/p \in \mathbb{Q} \cap (0,4)$, where $p$ is prime and $\gcd(r,p)=1$, we initiate the study of the algebraic…

Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of nonabelian simple groups. The minimum number of nonsolvable factors attained on all possible such series is called the…

Given a free group $F_k$ of rank $k\ge 2$ with a fixed set of free generators we associate to any homomorphism $\phi$ from $F_k$ to a group $G$ with a left-invariant semi-norm a generic stretching factor, $\lambda(\phi)$, which is a…

For $\alpha \in \mathbb{R}$, let $$G_{\alpha}:= \left< \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} , \begin{bmatrix} 1 & 0 \\ \alpha & 1 \end{bmatrix} \right> < \mathrm{SL}_2 (\mathbb{R}).$$ K. Kim and the first author established the…

Every finite group $G$ has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length $\lambda (G)$ as the minimum number of nonsoluble factors in a series of…

The $n$-dimensional quantum torus $\Lambda$ is defined to be the $F$-algebra generated by variables $y_1, \cdots, y_n$ with the relations $y_iy_j = q_{ij}y_jy_i$ where $q_{ij}$ are suitable scalars from the base field. This algebra is also…

A subset $A$ of a given finite abelian group $G$ is called $(k,l)$-sum-free if the sum of $k$ (not necessarily distinct) elements of $A$ does not equal the sum of $l$ (not necessarily distinct) elements of $A$. We are interested in finding…

An $L(2,1)$-labelling of a finite graph $\Gamma$ is a function that assigns integer values to the vertices $V(\Gamma)$ of $\Gamma$ (colouring of $V(\Gamma)$ by ${\mathbb{Z}}$) so that the absolute difference of two such values is at least…

We count the ordered sum-free triplets of subsets in the group $\mathbb{Z}/p\mathbb{Z}$, i.e., the triplets $(A,B,C)$ of sets $A,B,C \subset \mathbb{Z}/p\mathbb{Z}$ for which the equation $a+b=c$ has no solution with $a\in A$, $b \in B$ and…

The power graph $\Gamma_G$ of a finite group $G$ is the graph with the vertex set $G$, where two distinct elements are adjacent if one is a power of the other. An $L(2, 1)$-labeling of a graph $\Gamma$ is an assignment of labels from…

We show that for any finite-rank free group $\Gamma$, any word-equation in one variable of length $n$ with constants in $\Gamma$ fails to be satisfied by some element of $\Gamma$ of word-length $O(\log (n))$. By a result of the first…